where \(X^m\)  is a matrix consisting of all input data points. Although this bound has no explicit dependence on the network width (the dimensions of \(W_j,j\in [d]\) ), it has an exponential dependence on the network depth \(d\) , even if \(M_F(j)\le 1, \forall j\in [d]\) , which is obviously not desirable. [1]} improves this exponential dependency to polynomial dependency based on a covering numbers argument. The authors show that
\(\hat{\mathcal {R}}_m*{\mathcal {F}*{X^m}} \sim \tilde{\mathcal {O}} *{ B\prod _{j=1}^d *{W_j} \sqrt{\frac{d^{3} }{ m}} },\) 
